be a generically injective holomorphic map. The pair \((V, \phi )\)  is called a Teichmüller curve if \(V\)  has finite hyperbolic area and \(\phi \)  is an isometry with respect to the Kobayashi metric on the domainOn \(\mathbb {H}\) , and thus on \(V\) , the Kobayashi metric is equivalent to the Poincaré metric normalized such that its curvature is -4, see page 20 of [1]} and section 1 of [2]}. and the codomain. In other words, a Teichmüller curve is an irreducible algebraic curve in \(\mathcal {M}_g\)  which is a totally geodesic sub manifold for the Kobayashi (=Teichmüller) metric.
